The small boundary property in products

TL;DR

This paper characterizes when a group action on a compact space has the small boundary property, showing it is preserved under products and establishing a key lemma related to the Urysohn lemma.

Contribution

It proves the equivalence of small boundary property preservation under products and minimal actions without finite orbits, introducing a small boundary Urysohn lemma.

Findings

Small boundary property is equivalent to its preservation under product actions.

For minimal actions with the small boundary property, the property is automatic in product spaces.

A small boundary version of the Urysohn lemma is developed.

Abstract

For a continuous action $G\curvearrowright X$ of a countable group on a compact metrizable space we show that the following are equivalent: (i) the action $G\curvearrowright X$ has the small boundary property and no finite orbits, (ii) for every continuous action $H\curvearrowright Y$ of a countable group on a compact metrizable space, the product action $G\times H\curvearrowright X\times Y$ has the small boundary property. In particular, (ii) is automatic when $G$ is infinite and the action $G\curvearrowright X$ is minimal and has the small boundary property. The argument relies on a small boundary version of the Urysohn lemma.